Counting and numerical attributes of figures
What this domain tests
Many items hide an arithmetic progression in a visual attribute. You must count reliably and then look for the sequence. The counting itself is the difficulty: overlaps, shared vertices and hidden segments are designed to make you miscount.
What to count
Sides and edges, vertices, enclosed regions, intersection points, separate line segments, distinct shapes, dots inside versus outside, right angles, curved elements, and shaded versus unshaded regions.
Useful facts
- For a connected planar drawing, V minus E plus F equals 2, where F includes the outer face. Enclosed regions therefore equal E minus V plus 1.
- n straight lines in general position (no parallels, no three through one point) divide the plane into 1 plus n(n+1)/2 regions, of which (n minus 1)(n minus 2)/2 are bounded.
- A convex polygon with all diagonals from one vertex contains n minus 2 triangles.
- A circle with k diameters through the centre has 2k regions.
- A convex pentagon with all five diagonals drawn is divided into 11 regions; a regular hexagon with its three long diagonals into 6.
Worked example
Four straight lines are drawn with no two parallel and no three passing through a single point. Count the regions. The first line creates 2 regions; each new line crosses all previous lines and is cut into one more piece than the number of previous lines, so the additions are 1, 2, 3 and 4. Total regions equal 1 plus 1 plus 2 plus 3 plus 4, which is 11, with 3 of them bounded. If a figure in the series shows 4, 7, 11 regions, the step sizes are 3 and 4, so the next step is 5 and the answer shows 16 regions.
Traps that cost marks
- Counting the unbounded outer region as enclosed.
- Counting a triple intersection as three points instead of one.
- Missing segments hidden under overlapping shapes that still count as drawn.
- Counting a straight line drawn through a vertex as two segments when the attribute is lines.
- Using n squared where the correct formula is n(n+1)/2 plus 1.